The homogenization of a passive ‘tracer’ in a flow with closed mean streamlines occurs in two stages: first, a rapid phase dominated by shear-augmented diffusion over a time ≈P1/3(L/U), where the Péclet number P=LU/κ (L,U and κ are lengthscale, velocity scale and diffusivity), in which initial values of the tracer are replaced by their (generalized) average about a streamline; second, a slow phase requiring the full diffusion time ≈ L2/κ. The diffusion problem for the second phase, where tracer isopleths are held to streamlines by shear diffusion, involves a generalized diffusivity which is proportional to κ, but exceeds it if the streamlines are not circular. Expressions are also given for flow fields that are oscillatory rather than steady.

Earlier ideas are combined to produce a systematic approach both to forming the bulk equations of motion of a dilute suspension and to calculating the overall hydrodynamic interactions between the suspended particles. Equations governing averaged field quantities are derived by taking ensemble averages of the conservation laws and constitutive relations. The bulk equations thus produced contain a term in which the averaging is performed holding one particle fixed. If now the same prescription is applied to fields averaged with one particle fixed, equations are produced containing a term averaged with two particles fixed, and so on up an infinite hierarchy. The hierarchy can be truncated in an asymptotic analysis for small particle concentrations.

This approach to the mechanics of suspensions is illustrated by applying it to three problems which have already been well studied by different methods. The problems concern the first effects of hydrodynamic interactions on the bulk stress and sedimentation velocity of a free suspension, and on the permeability of a fixed bed. Earlier results are recovered in a new light. Multiparticle effects, which before have occurred as divergent sums, are seen to arise because the suspension described by the averaged equations assumes a viscosity and density different from the solvent, or in the case of the fixed bed because the suspension starts behaving as a porous medium instead of as a Newtonian solvent. A close connexion is thus revealed between the averaged-equation description of the interactions and a self-consistent-field model.

Steady-state two-phase flow in porous media was studied experimentally, using a model pore network of the chamber-and-throat type, etched in glass. The size of the network was sufficient to make end effects negligible. The capillary number, Ca, the flow-rate ratio, r, and the viscosity ratio, k, were changed systematically in a range that is of practical interest, whereas the wettability (moderate), the coalescence factor (high), and the geometrical and topological parameters of the porous medium were kept constant. Optical observations and macroscopic measurements were used to determine the flow regimes, and to calculate the corresponding relative permeabilities and fractional flow values. Four main flow regimes were observed and videorecorded, namely large-ganglion dynamics (LGD), small-ganglion dynamics (SGD), drop-traffic flow (DTF) and connected pathway flow (CPF). A map of the flow regimes is given in figure 3. The experimental demonstration that LGD, SGD and DTF prevail under flow conditions of practical interest, for which the widely held dogma presumes connected pathway flow, necessitates the drastic modification of that assumption. This is bound to have profound implications for the mathematical analysis and computer simulation of the process. The relative permeabilities are shown to correlate strongly with the flow regimes, figure 11. The relative permeability to oil (non-wetting fluid), kro, is minimal in the domain of LGD, and increases strongly as the flow mechanism changes from LGD to SGD to DTF to CPF. The relative permeability to water (wetting fluid), krw, is minimal in the domain of SGD; it increases moderately as the flow mechanism changes from SGD to LGD, whereas it increases strongly as the mechanism changes from SGD to DTF to CPF. Qualitative mechanistic explanations for these experimental results are proposed. The conventional relative permeabilities and the fractional flow of water, fw, are found to be strong functions not only of the water saturation, Sw, but also of Ca and k (with the wettability, the coalescence factor, and all the other parameters kept constant). These results imply that a fundamental reconsideration of fractional flow theory is warranted.

The non-equilibrium behaviour of concentrated colloidal dispersions is studied by Stokesian Dynamics, a general molecular-dynamics-like technique for simulating particles suspended in a viscous fluid. The simulations are of a suspension of monodisperse Brownian hard spheres in simple shear flow as a function of the Péclet number, Pe, which measures the relative importance of shear and Brownian forces. Three clearly defined regions of behaviour are revealed. There is first a Brownian-motion-dominated regime (Pe ≤ 1) where departures from equilibrium in structure and diffusion are small, but the suspension viscosity shear thins dramatically. When the Brownian and hydrodynamic forces balance (Pe ≈ 10), the dispersion forms a new ‘phase’ with the particles aligned in ‘strings’ along the flow direction and the strings are arranged hexagonally. This flow-induced ordering persists over a range of Pe and, while the structure and diffusivity now vary considerably, the rheology remains unchanged. Finally, there is a hydrodynamically dominated regime (Pe > 200) with a dramatic change in the long-time self-diffusivity and the rheology. Here, as the Péclet number increases the suspension shear thickens owing to the formation of large clusters. The simulation results are shown to agree well with experiment.

Measurements of the noise field from a 25 mm diameter subsonic air jet are presented. These results are analysed in some detail by determining both the jet velocity dependence and the directivity of the intensity of the radiation in 1/3-octave bands at particular values of the frequency parameter, \[ (fD/V_J)(1-M_c\cos\theta). \] This procedure should ensure that a particular source in a geometrically similar position in the jet is always observed, whatever the jet velocity, diameter and emission angle.

These results are compared with the predictions of Lighthil's (1952) theory of convected quadrupoles. It is shown that the theory predicts the variation of the intensity with jet velocity and emission angle provided that the observed frequency is below a certain critical value, which depends on jet diameter and emission angle and is independent of jet velocity. Above this critical frequency, the predicted variations overestimate the measurements and it appears that the convective amplification predicted by the theory is much reduced. The variation of this critical frequency is explained by assuming that substantial interaction occurs between the radiated sound and the jet flow when the wavelength of the sound becomes shorter than the sound path length in the jet flow.

The lateral migration of a solid spherical particle suspended in a fluid flowing between parallel vertical walls is investigated theoretically using a method developed by Cox & Brenner (1968). Buoyant and neutrally buoyant, freely rotating and non-rotating particles in the fluid flow are considered as is also the case of a sedimenting particle in a quiescent fluid. The results obtained are applied to the special cases of plane Poiseuille flow and of plane shear flow, these situations being investigated in detail.

The flow of a stratified fluid over topography is considered in the long-wavelength weakly nonlinear limit for the case when the flow is near resonance; that is, the basic flow speed is close to a linear long-wave phase speed for one of the long-wave modes. It is shown that the amplitude of this mode is governed by a forced Korteweg-de Vries equation. This equation is discussed both analytically and numerically for a variety of different cases, covering subcritical and supercritical flow, resonant or non-resonant, and for localized forcing that has either the same, or opposite, polarity to the solitary waves that would exist in the absence of forcing. In many cases a significant upstream disturbance is generated which consists of a train of solitary waves. The usefulness of internal hydraulic theory in interpreting the results is also demonstrated.

We discuss the creation of a finite-element program suitable for solving incompressible, viscous free-surface problems in steady axisymmetric or plane flows. For convenience in extending program capability to non-Newtonian flow, non-zero Reynolds numbers, and transient flow, a Galerkin formulation of the governing equations is chosen, rather than an extremum principle. The resulting program is used to solve the Newtonian die-swell problem for creeping jets free of surface tension constraints. We conclude that a Newtonian jet expands about 13%, in substantial agreement with experiments made with both small finite Reynolds numbers and small ratios of surface tension to viscous forces. The solutions to the related ‘stick-slip’ problem and the tube inlet problem, both of which also contain stress singularities, are also given.

From the conditions of energy and momentum conservation it is shown that all four interaction coefficients of the elementary quadruple interactions discussed in Part I of this paper are equal. A further conservative quantity is then found which can be interpreted as the number density of the gravity-wave ensemble representing the random sea surface. The well-known analogy between a random linear wave field and a mass particle ensemble is found to hold also in the case of weak non-linear interactions in the wave field. The interaction conditions for energy transfer between waves to correspond to the equations of energy and momentum conservation for a particle collision, the final equation for the rate of change of the wave spectrum corresponding to Boltzmann's equation for the rate of change of the number density of an ensemble of colliding mass particles. The stationary wave spectrum corresponding to the Maxwell distribution for a mass particle ensemble is found to be degenerate. The question of the irreversibility of the transfer process for gravity waves is discussed but not completely resolved.

An experimental investigation of the flow and acoustic properties of a moderate-Reynolds-number (Re = 70000), Mach number M = 2·1, axisymmetric jet has been performed. These measurements extended the experimental studies conducted previously in this laboratory to a higher-Reynolds-number regime where the flow and acoustic processes are considerably more complex. In fact, mean-flow and acoustic properties of this jet were determined to be closely comparable to published properties of high-Reynolds-number jets.

The major results of the flow-field measurements demonstrate that the jet shear annulus is unstable over a broad frequency range. The initial growth rates and wavelengths of these instabilities as measured by a hot wire were found to be in reasonable agreement with linear stability theory predictions. Also, in agreement with subsonic-jet results, the potential core of the jet was found to be most responsive to excitation at frequencies near a Strouhal number of S = 0·3. The overall development of organized disturbances around S = 0·2 seems to agree in general with calculations performed using the instability theory originally developed by Morris and Tam.

The acoustic near field was characterized in terms of sound-pressure level and directivity for both natural and excited (pure-tone) jets. In addition, propagation direction and azimuthal character of dominant spectral components were also measured. It was determined that the large-scale flow disturbances radiate noise in a directional pattern centred about 30° from the jet axis. The noise from these disturbances appears from simple ray tracing to be generated primarily near the region of the jet where the coherent fluctuations saturate in amplitude and begin to decay. It was also determined that the large-scale components of the near-field sound are made up predominately of axisymmetric (n = 0) and helical (n = ±1) modes. The dominant noise-generation mechanism appears to be a combination of Mach-wave generation and a process associated with the saturation and disintegration of the large-scale instability. Finally, the further development of a noise-generation model of the instability type appears to hold considerable promise.

The hydrodynamic stability of flows over Kramer-type compliant surfaces is studied. Two main types of instability are considered. First, there are those which could not exist without viscosity, termed Tollmien–Schlichting Type Instabilities (TSI). Secondly, there are Flow-Induced Surface Instabilities (FISI), that depend fundamentally on surface flexibility and could exist with an inviscid fluid flow. Part 1, the present paper, deals mainly with the first type. The original Kramer experiments and the various subsequent attempts to confirm his results are reviewed, together with experimental studies of transition in flows over compliant surfaces and theoretical work concerned with the hydrodynamic stability of such flows.

The Kramer-type compliant surface is assumed to be an elastic plate supported by springs which are modelled by an elastic foundation. It is also assumed that the plate is backed by a viscous fluid substrate having, in general, a density and viscosity different from the mainstream fluid. The motion of the substrate fluid is assumed to be unaffected by the presence of the springs and is determined by solving the linearized Navier–Stokes equations. The visco-elastic properties of the plate and springs are taken into account approximately by using a complex elastic modulus which leads to complex flexural rigidity and spring stiffness. Values for the various parameters characterizing the surface properties are estimated for the actual Kramer coatings.

The boundary-layer stability problem for a flexible surface is formulated in a similar way to that of Landahl (1962) whereby the boundary condition at the surface is expressed in terms of an equality between the surface and boundary-layer admittances. This form of the boundary condition is exploited to develop an approximate theory which determines whether a particular change to the mechanical properties of the surface will be stabilizing or destabilizing with respect to the TSI. It is shown that a reduction in flexural rigidity and spring stiffness, an increase in plate mass, and the presence of an inviscid fluid substrate are all stabilizing, whereas viscous and visco-elastic damping are destabilizing.

Numerical solutions to the Orr–Sommerfeld equation are also obtained. Apart from Kramer-type compliant surfaces, solutions are also presented for the rigid wall, for the spring-backed tensioned membrane with damping, previously considered by Landahl & Kaplan (1965), and for some of the compliant surfaces investigated experimentally by Babenko and his colleagues. The results for the Kramer-type compliant surfaces on the whole confirm the predictions of the simple theory. For a free-stream speed of 18 m/s the introduction of a viscous substrate leads to a complex modal interaction between the TSI and FISI. A single combined unstable mode is formed in the case of highly viscous substrate fluids and in this case increased damping has a stabilizing effect. When the free-stream speed is reduced to 15 m/s the modal interaction no longer occurs. In this case the effects of combined viscous and visco-elastic damping are investigated. It is found that damping tends to have a strong stabilizing effect on the FISI, in the form of travelling-wave flutter, but a weaker destabilizing effect on the TSI. The opposing effects of damping on the two modes of instability forms the basis of a possible explanation for Kramer's empirical observation of an optimum substrate viscosity. Results obtained using the e9 method also indicate that a substantial transition delay is theoretically possible for flows over Kramer's compliant coatings.

The single-domain Darcy–Brinkman model is applied to some analytically tractable flows through adjacent porous and pure-fluid domains and is compared systematically with the multiple-domain Stokes–Darcy model. In particular, we focus on the interaction between flow and solidification within the mushy layer during binary alloy solidification in a corner flow and on the effects of the chosen mathematical description on the resulting macrosegregation patterns. Large-scale results provided by the multiple-domain formulation depend strongly on the microscopic interfacial conditions. No satisfactory agreement between the single- and multiple-domain approaches is obtained when using previously suggested conditions written directly at the interface between the liquid and the porous medium. Rather, we define a viscous transition zone inside the porous domain, where the Stokes equation still applies, and we impose continuity of pressure and velocities across it. This new condition provides good agreement between the two formulations of solidification problems when there is a continuous variation of porosity across the interface between a partially solidified region (mushy zone) and the melt.

This paper studies the trajectories and near field of round jets in crossflow. Incompressible direct numerical simulations are performed at velocity ratios of 1.5 and 5.7 and the effects of jet velocity profile and boundary layer thickness on the jet trajectory are examined. The ‘rd’ scaling used at present (Margason 1993) does not contain any information on these parameters, and trajectories scaled by rd do not collapse. The trajectory is strongly influenced by the near field which depends on both the jet velocity profile and the crossflow boundary layer. A length scale is proposed to describe the near field of the jet. An analytical expression is proposed for this length scale which is a measure of the relative inertia of the jet and the crossflow. Incorporating this length scale significantly improves the scaling of the trajectories.

By using a Reynolds-averaged two-dimensional computation of a turbulent flow over an airfoil at post-stall angles of attack, we show that the massively separated and disordered unsteady flow can be effectively controlled by periodic blowing–suction near the leading edge with low-level power input. This unsteady forcing can modulate the evolution of the separated shear layer to promote the formation of concentrated lifting vortices, which in turn interact with trailing-edge vortices in a favourable manner and thereby alter the global deep-stall flow field. In a certain range of post-stall angles of attack and forcing frequencies, the unforced random separated flow can become periodic or quasi-periodic, associated with a significant lift enhancement. This opens a promising possibility for flight beyond the static stall to a much higher angle of attack. The same local control also leads, in some situations, to a reduction of drag. On a part of the airfoil the pressure fluctuation is suppressed as well, which would be beneficial for high-α buffet control. The computations are in qualitative agreement with several recent post-stall flow control experiments. The physical mechanisms responsible for post-stall flow control, as observed from the numerical data, are explored in terms of nonlinear mode competition and resonance, as well as vortex dynamics. The leading-edge shear layer and vortex shedding from the trailing edge are two basic constituents of unsteady post-stall flow and its control. Since the former has a rich spectrum of response to various disturbances, in a quite wide range the natural frequency of both constituents can shift and lock-in to the forcing frequency or its harmonics. Thus, most of the separated flow becomes resonant, associated with much more organized flow patterns. During this nonlinear process the coalescence of small vortices from the disturbed leading-edge shear layer is enhanced, causing a stronger entrainment and hence a stronger lifting vortex. Meanwhile, the unfavourable trailing-edge vortex is pushed downstream. The wake pattern also has a corresponding change: the shed vortices of alternate signs tend to be aligned, forming a train of close vortex couples with stronger downwash, rather than a Kármán street.

Stokes’ infinitesimal-wave expansion for steady progressive free-surface waves has been extended to high order using a computer to perform the coefficient arithmetic. Stokes’ expansion has been found to be incapable of yielding the highest wave for any value of the water depth since convergence is limited by a square-root branch-point some distance short of the maximum. By reformulating the problem using a different independent parameter, the highest waves are obtained correctly. Series summation and analytic continuation are facilitated by the use of Padé approximants. The method is valid in principle for any finite value of the wavelength and solutions of high accuracy can be obtained for most values of the wave height and water depth. An alternative expansion procedure proposed by Havelock for the computation of waves short of the highest has been reconsidered and found to be defective.

An experimental study of surface waves parametrically excited by vertical vibrations is presented. The shape of the eigenmodes in a closed vessel, and the importance of the free-surface boundary conditions, are discussed. Stability boundaries, wave amplitude, and perturbation characteristic time of decay are measured and found to be in agreement with an amplitude equation derived by symmetry. The measurement of the amplitude equation coefficients explains why the observed transition is always supercritical, and shows the effect of the edge constraint on the dissipation and eigen frequency of the various modes. The fluid surface tension is obtained from the dispersion relation measurement. Several visualization methods in large-aspect-ratio cells are presented.

Motivated by recent experimental and numerical studies of coherent structures in wall-bounded shear flows, we initiate a systematic exploration of the hierarchy of unstable invariant solutions of the Navier–Stokes equations. We construct a dynamical 105-dimensional state-space representation of plane Couette flow at Reynolds number Re = 400 in a small periodic cell and offer a new method of visualizing invariant manifolds embedded in such high dimensions. We compute a new equilibrium solution of plane Couette flow and the leading eigenvalues and eigenfunctions of known equilibria at this Re and cell size. What emerges from global continuations of their unstable manifolds is a surprisingly elegant dynamical-systems visualization of moderate-Re turbulence. The invariant manifolds partially tessellate the region of state space explored by transiently turbulent dynamics with a rigid web of symmetry-induced heteroclinic connections.

Large-eddy simulation (LES) was used to study mixing of turbulent, coannular jets discharging into a sudden expansion. This geometry resembles that of a coaxial jet-combustor, and the goal of the calculation was to gain some insight into the phenomena leading to lean blow-out (LBO) in such combustion devices. This is a first step in a series of calculations, where the focus is on the fluid dynamical aspects of the mixing process in the combustion chamber. The effects of swirl, chemical reactions and heat release were not taken into account. Mixing of fuel and oxidizer was studied by tracking a passive scalar introduced in the central jet. The dynamic subgrid-scale (DM) model was used to model both the subgrid-scale stresses and the subgrid-scale scalar flux. The Reynolds number was 38000, based on the bulk velocity and diameter of the combustion chamber. Mean velocities and Reynolds stresses are in good agreement with experimental data. Animated results clearly show that intermittent pockets of fuel-rich fluid (from the central jet) are able to cross the annular jet, virtually undiluted, into the recirculation zone. Most of the fuel-rich fluid is, however, entrained into the recirculation zone near the instantaneous reattachment point. Fuel trapped in the recirculation zone is, for the most part, entrained back into the step shear layer close to the base of the burner.

Some of the dense fluid at the front of an advancing gravity current is observed to be mixed with the ambient fluid. This process continues when the cross-stream non-uniformities at the head of the current are suppressed by advancing the floor beneath the head. In the resulting two-dimensional flow regular billows are visible. This paper considers experimentally and analytically the inviscid gravity current head and specifically includes the observed mixing at the head. Experimental results were obtained with an apparatus in which the head of the gravity current was brought to rest by an opposing uniform flow. The mixing appears to occur through Kelvin-Helmholtz billows generated on the front of the head and controls the dynamics of the head. A momentum balance is used to analyse the flow and the problem is closed by quantitatively introducing the billow structure.

Following ideas developed in the field of hydrodynamic stability of laminar flows (Stuart 1971) a predictive theory is proposed to determine the development of finite-amplitude alternate bars in straight channels with erodible bottoms. It is shown that an ‘equilibrium amplitude’ of bedforms is reached as t → ∞ within a wide range of values of the parameter (β − βc)/βc, where t is the time, β is the width ratio of the channel and βc is its ‘critical’ value below which bars would not form. The theory leads to relationships for the maximum height and the maximum scour of bars which compare satisfactorily with the experimental data of various authors. Moreover the experimentally detected tendency of the bed perturbation to form diagonal fronts is qualitatively reproduced.